Uniform Pointwise Convergence of Finite Difference Schemes Using Grid Equidistribution

Abstract

A singularly perturbed quasilinear two-point boundary value problem is considered. The problem is discretized using a simple upwind finite difference scheme on adapted meshes using grid equidistribution of monitor functions. We derive sufficient conditions on the monitor function that guarantee uniform convergence in the discrete maximum norm no matter how small the perturbation parameter is. These results can be used to deduce uniform convergence of the scheme for a number of layer-adapted meshes. We also propose an adaptive procedure for the numerical treatment of the boundary value problem. Numerical experiments for the schemes are presented.